A wide range of macromolecules can undergo phase separation, forming biomolecular condensates in living cells. These membraneless organelles are typically highly dynamic, formed reversibly, and carry out essential functions in biological systems. Crucially, however, a further liquid-to-solid transition of the condensates can lead to irreversible pathological aggregation and cellular dysfunction associated with the onset and development of neurodegenerative diseases. Despite the importance of this liquid-to-solid transition of proteins, the mechanism by which it is initiated in normally functional condensates is unknown. Here we show, by measuring the changes in structure, dynamics and mechanics in time and space, that single component FUS condensates do not uniformly convert to a solid gel, but rather that liquid and gel phases co-exist simultaneously within the same condensate, resulting in highly inhomogeneous structures, and that, crucially, this transition originates at the interface and propagates towards the center of the condensate. We introduce two new optical techniques, Spatial Dynamic Mapping and Reflective Confocal Dynamic Speckle Microscopy. We use these to further monitor and quantify the local dynamics of the liquid-to-solid transition and demonstrate that is initiated at the interface between the dense phase within condensates and the dilute phase. These results reveal the importance of the spatiotemporal dimension of the liquid-to-solid transition and highlight the interface of biomolecular condensates as a critical element in driving pathological protein aggregation.